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The discrete-time quantum walk (QW) is a quantum version of the random walk(RW) and has been widely investigated for the last two decades. Some remarkableproperties of QW are well known. For example, QW has a ballistic spreading,i.e., QW is quadratically faster than RW. For some cases, localization occurs:a walker stays at the starting position forever. In this paper, we considerstationary measures of two-state QWs on the line. It was shown that for anyspace-homogeneous model, the uniform measure becomes the stationary measure.However, the corresponding result for space-inhomogeneous model is not known.Here, we present a class of space-inhomogeneous QWs on the line and cycles inwhich the uniform measure is stationary. Furthermore, we briefly discuss adifference between QWs and RWs.

Uniform stationary measure of space-inhomogeneous quantum walks in one dimension